Answer:
A) The average velocity is 144 m/s.
B) The average velocity is (111 + h) m/s.
C) The instantaneous velocity is 111 m/s.
Explanation:
The position of the object is given by the following function:
y = f(x) = x² + x
A) The average velocity can be calculated as follows:
av = (f(xf) - f(x0)) / (xf - x0)
Where:
av = average velocity
f(xf) = the value of the function at x = x-final (xf)
f(x0) = the value of the function at x = x-initial (x0)
xf = x-final, final value of "x"
x0 = x-initial, initial value of "x"
Then the averge rate of change from x0 = 55 to xf =88 will be:
av = (f(88) - f(55))/(88 - 55)
f(x) = x² + x
Evaluating the function in x = 88 and x = 55:
f(88) = 88² + 88 = 7832 m
f(55) = 55² + 55 = 3080 m
Then:
av = 7832 m - 3080 m / (88 s - 55 s) =<u> 144 m/s</u>
B) av = (f(55 + h) - f(55)) / (55 + h - 55)
Evaluating the function in x = 55 +h:
f(55+h) = (55+h)² + (55+h)
f(55+h) = (55+h) · (55+h) + (55 + h)
f(55+h) = 55² + 55h +55h + h² + 55 +h
Evaluating the function in x = 55
f(55) = 55² + 55
Then f(xf) - f(x0):
f(55+h) - f(55) = 55² + 55h +55h + h² + 55 +h - 55² - 55
f(55+h) - f(55) = h(111 + h)
Then:
av = h(111 + h) / h = (<u>111 + h) m/s</u> (if h ≠ 0)
C) The instantaneous velocity is obtained by derivating the function and evaluating the derivative at x = 55.
Then:
f´(x) = 2x + 1
f´(55) = 2 · 55 + 1 = <u>111 m/s</u>
